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September  2018, 17(5): 2085-2123. doi: 10.3934/cpaa.2018099

Concentration phenomena for critical fractional Schrödinger systems

Vincenzo Ambrosio

Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino 'Carlo Bo', Piazza della Repubblica, 13 61029 Urbino (Pesaro e Urbino), Italy

Received  October 2017 Revised  November 2017 Published  April 2018

In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schrödinger system
$\left\{ \begin{array}{*{35}{l}} \begin{align} & {{\varepsilon }^{2s}}{{(-\Delta )}^{s}}u+V(x)u={{Q}_{u}}(u,v)+\frac{1}{2_{s}^{*}}{{K}_{u}}(u,v)\ \ \ \ \ \text{in }{{\mathbb{R}}^{N}} \\ & {{\varepsilon }^{2s}}{{(-\Delta )}^{s}}u+W(x)v={{Q}_{v}}(u,v)+\frac{1}{2_{s}^{*}}{{K}_{v}}(u,v)\ \ \ \ \text{in }{{\mathbb{R}}^{N}} \\ & u,v>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ in }{{\mathbb{R}}^{N}}, \\ \end{align} & \text{ } & \text{ } & {} \\\end{array} \right.$
where
$\varepsilon>0$
 is a parameter,
$s∈ (0, 1)$
,
$N>2s$
,
$(-Δ)^{s}$
 is the fractional Laplacian operator,
$V:\mathbb{R}^{N} \to \mathbb{R}$
 and
$W:\mathbb{R}^{N} \to \mathbb{R}$
 are positive Hölder continuous potentials,
$Q$
 and
$K$
 are homogeneous
$C^{2}$
-functions having subcritical and critical growth respectively.
We relate the number of solutions with the topology of the set where the potentials
$V$
 and
$W$
 attain their minimum values. The proofs rely on the Ljusternik-Schnirelmann theory and variational methods.
Keywords: Fractional Schrödinger systems, variational methods, critical exponent.
Mathematics Subject Classification: Primary: 35A15, 35J50; Secondary: 35R11, 58E05.
Citation: Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099
References:
[1]

C. O. Alves, Local mountain pass for a class of elliptic system, J. Math. Anal. Appl., 335 (2007), 135-150. 

[2]

C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., Ser. A: Theory Methods, 42 (2000), 771-787. 

[3]

C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiplicity of solutions for elliptic systems via local mountain pass method, Commun. Pure Appl. Anal., 8 (2009), 1745-1758. 

[4]

C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for critical elliptic systems via penalization method, Differential Integral Equations, 23 (2010), 703-723. 

[5]

C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp.

[6]

C. O. Alves and S. H. M. Soares, Existence and concentration of positive solutions for a class of gradient systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 437-457. 

[7]

V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp.

[8]

V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, accepted for publication in Rev. Mat. Iberoamericana, (arXiv: 1612.02388).

[9]

V. Ambrosio, Multiplicity of solutions for fractional Schrödinger systems in $\mathbb{R}^{N}$, preprint arXiv: 1703.04370.

[10]

V. Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl., 196 (2017), 2043-2062. 

[11]

V. Ambrosio, Multiplicity and concentration of solutions for fractional Schrödinger systems via penalization method, preprint arXiv: 1703.04370.

[12]

V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), no. 2,615-645. 

[13]

D. Applebaum, Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. xxx+460 pp.

[14]

A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 459-479. 

[15]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48. 

[16]

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. 

[17]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.

[18]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. 

[19]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. 

[20]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. 

[21]

W. Choi, On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal., 120 (2015), 127-153. 

[22]

J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. 

[23]

J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892. 

[24]

D. C. de Morais Filho and M. A. S. Souto, Systems of $p$ -Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees, Comm. Partial Differential Equations, 24 (1999), 1537-1553. 

[25]

M. Del Pino and P. L. Felmer, Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. 

[26]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. 

[27]

S. Dipierro, M. Medina, I. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb{R}^{N}$, Manuscripta Math., 153 (2017), 183-230. 

[28]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp.

[29]

S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differential Equations, 255 (2013), 85-119. 

[30]

L. F. O. Faria, O. H. Miyagaki, F. R. Pereira, M. Squassina and C. Zhang, The Brezis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2016), 85-103. 

[31]

P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. 

[32]

G. M. Figueiredo and M. F. Furtado, Multiple positive solutions for a quasilinear system of Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 309-333. 

[33]

A. Fiscella and P. Pucci, $p$ -fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378. 

[34]

Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706. 

[35]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 91, 39 pp.

[36]

H. Hajaiej, Symmetric ground states solutions of m-coupled nonlinear Schrödinger equations, Nonlinear Anal., 71 (2009), 4696-4704. 

[37]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. 

[38]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108.

[39]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Part $I$., Rev. Mat. Iberoamericana, 1 (1985), 145-201. 

[40]

B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135. 

[41]

G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, with a foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. xvi+383 pp.

[42]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829. 

[43]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, J. Math. Phys., 54 (2013), 031501.

[44]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. 

[45]

X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207. 

[46]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. 

[47]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J., 1970.

[48]

K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106. 

[49]

S. Terracini, G. Verzini and A. Zilio, Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian, J. Eur. Math. Soc. (JEMS), 18 (2016), 2865-2924. 

[50]

Y. Wan and A. Ávila, Multiple solutions of a coupled nonlinear Schrödinger system, J. Math. Anal. Appl., 334 (2007), 1308-1325

[51]

K. Wang and J. Wei, On the uniqueness of solutions of a nonlocal elliptic system, Math. Ann., 365 (2016), 105-153. 

[52]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. x+162 pp.

[53]

Z. Xia, B. Zhang and D. Repovs, Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal., 142 (2016), 48-68. 

show all references

References:
[1]

C. O. Alves, Local mountain pass for a class of elliptic system, J. Math. Anal. Appl., 335 (2007), 135-150. 

[2]

C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., Ser. A: Theory Methods, 42 (2000), 771-787. 

[3]

C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiplicity of solutions for elliptic systems via local mountain pass method, Commun. Pure Appl. Anal., 8 (2009), 1745-1758. 

[4]

C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for critical elliptic systems via penalization method, Differential Integral Equations, 23 (2010), 703-723. 

[5]

C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), Art. 47, 19 pp.

[6]

C. O. Alves and S. H. M. Soares, Existence and concentration of positive solutions for a class of gradient systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 437-457. 

[7]

V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp.

[8]

V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, accepted for publication in Rev. Mat. Iberoamericana, (arXiv: 1612.02388).

[9]

V. Ambrosio, Multiplicity of solutions for fractional Schrödinger systems in $\mathbb{R}^{N}$, preprint arXiv: 1703.04370.

[10]

V. Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl., 196 (2017), 2043-2062. 

[11]

V. Ambrosio, Multiplicity and concentration of solutions for fractional Schrödinger systems via penalization method, preprint arXiv: 1703.04370.

[12]

V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation, Math. Methods Appl. Sci., 41 (2018), no. 2,615-645. 

[13]

D. Applebaum, Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. xxx+460 pp.

[14]

A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 12 (2005), 459-479. 

[15]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48. 

[16]

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. 

[17]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.

[18]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. 

[19]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. 

[20]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. 

[21]

W. Choi, On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal., 120 (2015), 127-153. 

[22]

J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. 

[23]

J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892. 

[24]

D. C. de Morais Filho and M. A. S. Souto, Systems of $p$ -Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees, Comm. Partial Differential Equations, 24 (1999), 1537-1553. 

[25]

M. Del Pino and P. L. Felmer, Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. 

[26]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. 

[27]

S. Dipierro, M. Medina, I. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb{R}^{N}$, Manuscripta Math., 153 (2017), 183-230. 

[28]

S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp.

[29]

S. Dipierro and A. Pinamonti, A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian, J. Differential Equations, 255 (2013), 85-119. 

[30]

L. F. O. Faria, O. H. Miyagaki, F. R. Pereira, M. Squassina and C. Zhang, The Brezis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2016), 85-103. 

[31]

P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. 

[32]

G. M. Figueiredo and M. F. Furtado, Multiple positive solutions for a quasilinear system of Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 309-333. 

[33]

A. Fiscella and P. Pucci, $p$ -fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378. 

[34]

Z. Guo, S. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706. 

[35]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 91, 39 pp.

[36]

H. Hajaiej, Symmetric ground states solutions of m-coupled nonlinear Schrödinger equations, Nonlinear Anal., 71 (2009), 4696-4704. 

[37]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. 

[38]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108.

[39]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Part $I$., Rev. Mat. Iberoamericana, 1 (1985), 145-201. 

[40]

B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal., 146 (2016), 120-135. 

[41]

G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, with a foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. xvi+383 pp.

[42]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829. 

[43]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, J. Math. Phys., 54 (2013), 031501.

[44]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. 

[45]

X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207. 

[46]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. 

[47]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N. J., 1970.

[48]

K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106. 

[49]

S. Terracini, G. Verzini and A. Zilio, Uniform Hölder bounds for strongly competing systems involving the square root of the Laplacian, J. Eur. Math. Soc. (JEMS), 18 (2016), 2865-2924. 

[50]

Y. Wan and A. Ávila, Multiple solutions of a coupled nonlinear Schrödinger system, J. Math. Anal. Appl., 334 (2007), 1308-1325

[51]

K. Wang and J. Wei, On the uniqueness of solutions of a nonlocal elliptic system, Math. Ann., 365 (2016), 105-153. 

[52]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. x+162 pp.

[53]

Z. Xia, B. Zhang and D. Repovs, Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal., 142 (2016), 48-68. 

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